To study the intrinsic physics and improve computation efficiency of free–surface flow problems, in this study we propose a proper orthogonal decomposition (POD) technique that couples the velocity flow field and the level–set function field for free–surface flows. In this method, the snapshots data from numerical or experimental results are used to assemble a low–dimensional basis so that the flow characteristics can be retrieved with a priori knowledge of equal distribution of the total variance between velocity and level–set function data. Through numerical examples of a sloshing problem and a water entry problem, we show that the low–dimensional components obtained provide an efficient and accurate approximation of the flow field Moreover, we show that the velocity contour and orbits projected on the space of the reduced basis greatly facilitate understanding of the intrinsic dynamics of the flow systems.

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